Holt McDougal Larson Pre-Algebra Teacher Solutions Key Worksheets ... - Free Printable

The image shows a page from a mathematics textbook, specifically focusing on Chapter 4: Factors and Prime Factorization. The right-hand side of the page is titled "4.1 Finding Prime Numbers", which introduces the concept of prime numbers and provides an activity to identify them using patterns. Below is a detailed explanation of the content and how to solve the problem presented.

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Key Concepts in the Image

1. Definition of Prime Numbers:
- A prime number is a whole number greater than 1 that has exactly two distinct positive divisors: 1 and itself.
- Examples: 2, 3, 5, 7, 11, etc.

2. Definition of Composite Numbers:
- A composite number is a whole number greater than 1 that has more than two distinct positive divisors.
- Examples: 4, 6, 8, 9, 10, etc.

3. Activity Instructions:
- Write the whole numbers from 2 to 60 in rows of 6.
- Use a systematic method (Sieve of Eratosthenes) to identify prime numbers by eliminating composite numbers.

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Step-by-Step Solution for the Activity

#### Step 1: Write the Whole Numbers from 2 to 60 in Rows of 6
The numbers are already provided in the image:
```
2 3 4 5 6 7
8 9 10 11 12 13
14 15 16 17 18 19
20 21 22 23 24 25
26 27 28 29 30 31
32 33 34 35 36 37
38 39 40 41 42 43
44 45 46 47 48 49
50 51 52 53 54 55
56 57 58 59 60
```

#### Step 2: Start with the Number 2
- Circle the number 2 because it is the smallest prime number.
- Cross out all multiples of 2 (i.e., even numbers): 4, 6, 8, 10, 12, ..., 60.

#### Step 3: Move to the Next Uncrossed Number
- The next uncrossed number is 3. Circle it because it is prime.
- Cross out all multiples of 3: 6, 9, 12, 15, 18, ..., 60. Note that some of these numbers (e.g., 6, 12, 18) are already crossed out as multiples of 2.

#### Step 4: Repeat the Process
- The next uncrossed number is 5. Circle it because it is prime.
- Cross out all multiples of 5: 10, 15, 20, 25, 30, ..., 60. Again, some of these numbers (e.g., 10, 15, 20) are already crossed out.

- The next uncrossed number is 7. Circle it because it is prime.
- Cross out all multiples of 7: 14, 21, 28, 35, 42, 49, 56. Some of these numbers (e.g., 14, 21, 28) are already crossed out.

- Continue this process until every number is either circled (prime) or crossed out (composite).

#### Final Result
After completing the steps, the circled numbers (primes) between 2 and 60 are:
```
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59
```

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Draw Conclusions

1. What can you say about the numbers that have been crossed out?
- These numbers are composite because they have more than two distinct positive divisors.

2. What can you say about the numbers that have been circled?
- These numbers are prime because they have exactly two distinct positive divisors: 1 and themselves.

3. Critical Thinking:
- All numbers in the sixth column are crossed out because they are multiples of 6. This is because any number in the sixth column is divisible by both 2 and 3, making it composite.
- Similarly, numbers in the second column (except 2), third column (except 3), and fourth column are crossed out because they are multiples of smaller primes.

4. Prediction:
- If the numbers were arranged in rows of 10 instead of 6, the columns containing multiples of 2, 3, 5, and 7 would be crossed out. Specifically:
- Column 2: Multiples of 2.
- Column 4: Multiples of 2.
- Column 6: Multiples of 2 and 3.
- Column 8: Multiples of 2.
- Column 10: Multiples of 2 and 5.

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Final Answer

The prime numbers between 2 and 60 are:
\[
\boxed{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59}
\]